it takes 100 pet foods to buy a pet and each start at level 1.
"If you give your Pet a special Item on this menu, it will also Level up (again, 100 max). Pets give special bonuses at max Level!"
There are 6 listed pets, and they are
Ninja Kitty lv 0 (1%-20% Kaiju damage)
Cave Puppy lv 0 (1%-20% Item Find)
Awkward Penguin lv 0 (???) you need 30 total levels in pets before you can get this one
Ninja Tortoise lv 0 (???) you need 30 total levels in pets before you can get this one
Honey Badger lv 0 (???) you need 100 total levels in pets before you can get this one
Ninja squirrel lv 0 (???) you need 100 total levels in pets before you can get this one